 
Summary: Online and Approximation Algorithms Fall Semester, 2004/2005
Exercise 4 (EXAM): January 12, 2005
Lecturer: Prof. Yossi Azar
Write short but full and accurate answers. Each question should start on a new page and should
not exceed a page (both sides).
1. We are given a connected graph G = (V, E), where V  = n. All edges have unit capacity. At
step i we receive a request (si, ti) of bandwidth precisely 1/3. The goal is to maximize the total
bandwidth of accepted requests (preemption is not allowed).
(a) Design an O(n1/3) competitive algorithm for the problem. Hint: use weights of about
0, n1/3, n2/3, n.
(b) How would you modify the algorithm and the analysis to achieve an O(kn1/3) competitive
algorithm if all requests are of bandwidth between 1/k and 1/3 for any k > 3.
2. Consider online load balancing of permanent tasks on m machines in the restricted assignment
model. Consider the case where for each i the set of machines which is associated with job i is
[1 . . . ki] for some 1 ki m.
(a) Design a constant competitive algorithm.
(b) Show a lower bound of 2 for deterministic algorithms, for any m 6, using only unit jobs.
(c) Show a lower bound of 2  O(1/m) for randomized algorithms, for any m, using only unit
jobs.
3. Suppose we are given one machine and a set of jobs that arrive over time. The machine can
